Nonlinear Vibration of Parametrically Excited, Viscoelastic, Axially Moving Strings

2005 ◽  
Vol 72 (3) ◽  
pp. 374-380 ◽  
Author(s):  
Eric M. Mockensturm ◽  
Jianping Guo
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Time Derivative ◽  
Excitation Frequency ◽  
Viscoelastic Behavior ◽  
Solution Procedure ◽  
Governing Equations ◽  
Translation Speed ◽  
Parametrically Excited ◽  
Axially Moving

The dynamic response of parametrically excited, axially moving viscoelastic belts is investigated in this paper. Results are compared to previous work in which the partial, not material, time derivative was used in the viscoelastic constitutive relation. It is found that this added “steady state” dissipation greatly affects both the existence and amplitudes of nontrivial limit cycles. The discrepancy increases with increasing translation speed. To limit the comparison to the additional physics included in the model, the solution procedure of Zhang and Zu [1,2], who applied the method of multiple scales to the governing equations prior to discretization, is retained. The excitation here is provided by physically stretching the belt. In this case, viscoelastic behavior and excitation frequency also affects the amplitude of the tension fluctuations.

A New Multi-Pulse Chaotic Motion for Parametrically Excited Viscoelastic Axially Moving String

2005 ◽  
Author(s):  
Wei Zhang ◽  
Ming-Hui Yao ◽  
Li-Lai Bai
Keyword(s):  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Three Dimensional ◽  
Constitutive Law ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Dimensional Phase Space ◽  
Axially Moving ◽  
First Time

In this paper, the Shilnikov type multi-pulse orbits and chaotic dynamics of parametrically excited viscoelastic moving string are studied in detail. Using Kelvin-type viscoelastic constitutive law, the equation of motion for viscoelastic moving string with the external damping and parametric excitation is given. The four-dimensional averaged equation under primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of viscoelastic moving string. The Shilnikov type multi-pulse chaotic motions of viscoelastic moving string are also found by using numerical simulation. A new phenomenon on the multi-pulse jumping orbits and a new strange attractor are observed from three-dimensional phase space for the first time.

Parametric Stability of Axially Accelerating Viscoelastic Beams With the Recognition of Longitudinally Varying Tensions

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Li-Qun Chen ◽  
You-Qi Tang
Keyword(s):  
Governing Equation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Finite Support ◽  
Governing Equations ◽  
Parametric Stability ◽  
Stability Boundaries ◽  
Axial Acceleration ◽  
The Stability ◽  
Longitudinally Varying Tension

In this paper, the parametric stability of axially accelerating viscoelastic beams is revisited. The effects of the longitudinally varying tension due to the axial acceleration are highlighted, while the tension was approximately assumed to be longitudinally uniform in previous studies. The dependence of the tension on the finite support rigidity is also considered. The generalized Hamilton principle and the Kelvin viscoelastic constitutive relation are applied to establish the governing equations and the associated boundary conditions for coupled planar motion of the beam. The governing equations are linearized into the governing equation in the transverse direction and the expression of the longitudinally varying tension. The method of multiple scales is employed to analyze the parametric stability of transverse motion. The stability boundaries are derived from the solvability conditions and the Routh-Hurwitz criterion for principal and sum resonances. In terms of stability boundaries, the governing equations with or without the longitudinal variance of tension are compared and the effects of the finite support rigidity are also examined. Some numerical examples are presented to demonstrate the effects of the stiffness, the viscosity, and the mean axial speed on the stability boundaries. The differential quadrature scheme is developed to numerically solve the governing equation, and the computational results confirm the outcomes of the method of multiple scales.

Response variations of a cantilever beam–tip mass system with nonlinear and linearized boundary conditions

2018 ◽  
Vol 25 (3) ◽  
pp. 485-496 ◽  
Author(s):  
Vamsi C. Meesala ◽  
Muhammad R. Hajj
Keyword(s):  
Boundary Conditions ◽  
Cantilever Beam ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Boundary Conditions ◽  
Distributed Parameter ◽  
Governing Equations ◽  
Mass System ◽  
Tip Mass

The distributed parameter governing equations of a cantilever beam with a tip mass subjected to principal parametric excitation are developed using a generalized Hamilton's principle. Using a Galerkin's discretization scheme, the discretized equation for the first mode is developed for simpler representation assuming linear and nonlinear boundary conditions. The discretized governing equation considering the nonlinear boundary conditions assumes a simpler form. We solve the distributed parameter and discretized equations separately using the method of multiple scales. Through comparison with the direct approach, we show that accounting for the nonlinear boundary conditions boundary conditions is important for accurate prediction in terms of type of bifurcation and response amplitude.

Primary Resonance Voltage Response of Electrostatically Actuated M/NEMS Circular Plate Resonators

2014 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Reynaldo Oyervides
Keyword(s):  
Casimir Force ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Voltage Response ◽  
Circular Plates ◽  
Amplitude Response ◽  
Weakly Nonlinear ◽  
Electrostatically Actuated

This paper investigates the voltage-amplitude response of soft AC electrostatically actuated M/NEMS clamped circular plates. AC frequency is near half natural frequency of the plate. This results in primary resonance. The system is analytically modeled using the Method of Multiple Scales (MMS). The system is assumed weakly nonlinear. The behavior of the system including pull-in instability as the AC voltage is swept up and down while the excitation frequency is constant is reported. The effects of detuning frequency, damping, Casimir force, and van der Waals force are reported as well.

scholarly journals Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification

2012 ◽  
Vol 19 (4) ◽  
pp. 527-543 ◽  
Author(s):  
Li-Qun Chen ◽  
Hu Ding ◽  
C.W. Lim
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Steady State ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Steady State Response ◽  
Governing Equations ◽  
State Response ◽  
Non Linear ◽  
Axial Speed

Transverse non-linear vibration is investigated in principal parametric resonance of an axially accelerating viscoelastic beam. The axial speed is characterized as a simple harmonic variation about a constant mean speed. The material time derivative is used in the viscoelastic constitutive relation. The transverse motion can be governed by a non-linear partial-differential equation or a non-linear integro-partial-differential equation. The method of multiple scales is applied to the governing equations to determine steady-state responses. It is confirmed that the mode uninvolved in the resonance has no effect on the steady-state response. The differential quadrature schemes are developed to verify results via the method of multiple scales. It is demonstrated that the straight equilibrium configuration becomes unstable and a stable steady-state emerges when the axial speed variation frequency is close to twice any linear natural frequency. The results derived for two governing equations are qualitatively the same, but quantitatively different. Numerical simulations are presented to examine the effects of the mean speed and the variation of the amplitude of the axial speed, the dynamic viscosity, the non-linear coefficients, and the boundary constraint stiffness on the instability interval and the steady-state response amplitude.

On 2D Forcespinning™ Modeling

2011 ◽  
Author(s):  
Simon Padron ◽  
Dumitru I. Caruntu ◽  
Karen Lozano
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Fiber Diameter ◽  
Production Method ◽  
Fiber Formation ◽  
Centrifugal Forces ◽  
Governing Equations ◽  
Novel Method ◽  
Jet Theory ◽  
High Yields

Forcespinning™ is a novel method that makes used of centrifugal forces to produce nanofibers rapidly and at high yields. To improve and enhance this new nanofiber production method a model of the system is begun. The process is started by deriving the governing equations of the forcespinning™ sytem and the constraints associated it. A simple 2D model is then obtained using the derived governing equations for the inviscid case to determine the trends of fiber diameter and trajectories. Then, focus is given to the time-dependency of these equations, and the effects of parametric excitation of the system on fiber formation are analyzed. The equations are solved using a combination of the method of multiple scales and the finite difference method with slender-jet theory assumptions.

Nonlinear Vibration of Parametrically Excited Viscoelastic Moving Belts

1999 ◽  
Author(s):  
Lixin Zhang ◽  
Jean W. Zu
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
Generalized Equation ◽  
Constitutive Law ◽  
Parametrically Excited ◽  
Viscoelastic Parameters ◽  
Linear Viscoelastic ◽  
Existence Conditions ◽  
Gyroscopic Systems

Abstract The dynamic response and stability of parametrically excited viscoelastic belts are investigated in this paper. The linear viscoelastic differential constitutive law is employed to characterize the material property of belts. The generalized equation of motion is obtained for a viscoelastic moving belt with geometric nonlinearity. The method of multiple scales is applied directly to the governing equation, which is in the form of continuous gyroscopic systems. Closed-form expressions for the amplitude, existence conditions and stability conditions of non-trivial limit cycles of the summation resonance are obtained. Effects of viscoelastic parameters, excitation frequencies, excitation amplitudes and axial moving speeds on stability boundaries are discussed.

Parametric Instability of Axially Moving Media Subjected to Multifrequency Tension and Speed Fluctuations

2000 ◽  
Vol 68 (1) ◽  
pp. 49-57 ◽  
Author(s):  
R. G. Parker ◽  
Y. Lin
Keyword(s):  
Multiple Scales ◽  
Parametric Instability ◽  
Translation Speed ◽  
Excitation Frequencies ◽  
Axially Moving ◽  
Frequency Changes ◽  
Limit Cycle Amplitude ◽  
Moving Media ◽  
Secondary Instabilities ◽  
Primary Instability

This work investigates the stability of axially moving media subjected to parametric excitation resulting from tension and translation speed oscillations. Each of these excitation sources has spectral content with multiple frequencies and arbitrary phases. Stability boundaries for primary parametric instabilities, secondary instabilities, and combination instabilities are determined analytically through second-order perturbation. The classical result that primary instability occurs when one of the excitation frequencies is close to twice a natural frequency changes as a result of multiple excitation frequencies. Unusual interactions occur for the practically important case of simultaneous primary and secondary instabilities. While sum type combination instabilities occur, no difference type instabilities are detected. The nonlinear limit cycle amplitude that occurs under primary instability is derived using the method of multiple scales.

Determining Stability Boundaries Using Gyroscopic Eigenfunctions

2003 ◽  
Vol 125 (3) ◽  
pp. 405-407 ◽  
Author(s):  
Anthony A. Renshaw
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Stability Boundaries ◽  
Parametrically Excited ◽  
The Stability ◽  
Simplified Procedure ◽  
Gyroscopic Systems

By taking advantage of modal decoupling and reduction of order, we derive a simplified procedure for applying the method of multiple scales to determine the stability boundaries of parametrically excited, gyroscopic systems. The analytic advantages of the procedure are illustrated with three examples.

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